Towards parameterizing the entanglement body of a qubit pair

Abstract

A method allowing to increase a computational efficiency of evaluation of non-local characteristics of a pair of qubits is described. The method is based on the construction of coordinates on a generic section of 2-qubit's entanglement space ${\mathcal{E}}_{2\times 2}$ represented as the direct product of an ordered 3-dimensional simplex and the double coset $\mathrm{SU}(2)\times \mathrm{SU}(2)\backslash \mathrm{SU}(4)/{\mathrm{T}}^3.$ Within this framework, the subset ${\mathcal{S}\mathcal{E}}_{2\times 2}\subset {\mathcal{E}}_{2\times 2}$ corresponding to the rank-4 separable 2-qubit states is described as a semialgebraic variety given by a system of 3rd and 4th order polynomial inequalities in eigenvalues of the density matrix, whereas the polynomials coefficients are trigonometric functions defined over a direct product of two regular octahedra.